Skip to main content
SpringerLink
Log in

Differential transformation method for studying flow and heat transfer due to stretching sheet embedded in porous medium with variable thickness, variable thermal conductivity, and thermal radiation

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

This article presents a numerical solution for the flow of a Newtonian fluid over an impermeable stretching sheet embedded in a porous medium with the power law surface velocity and variable thickness in the presence of thermal radiation. The flow is caused by non-linear stretching of a sheet. Thermal conductivity of the fluid is assumed to vary linearly with temperature. The governing partial differential equations (PDEs) are transformed into a system of coupled non-linear ordinary differential equations (ODEs) with appropriate boundary conditions for various physical parameters. The remaining system of ODEs is solved numerically using a differential transformation method (DTM). The effects of the porous parameter, the wall thickness parameter, the radiation parameter, the thermal conductivity parameter, and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction and the Nusselt numbers are presented. Comparison of the obtained numerical results is made with previously published results in some special cases, with good agreement. The results obtained in this paper confirm the idea that DTM is a powerful mathematical tool and can be applied to a large class of linear and non-linear problems in different fields of science and engineering.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

A, b :

constants

C f :

skin friction coefficient

c p :

specific heat at constant pressure

D :

porous parameter

f(η):

similarity variable

k :

permeability of porous medium

k*:

mean absorption coefficient

k 0 :

permeability parameter

m :

velocity power index

Nu :

Nusselt number

Pr :

Prandtl number

q r :

radiative heat flux

q eff :

effective conduction-radiation flux

R :

radiation parameter

Re x :

local Reynolds number

T :

temperature of fluid

T w :

temperature of sheet

T ∞ :

free-stream temperature

u :

velocity component in x-direction

v :

velocity component in y-direction

U :

stretching velocity

U 0 :

reference velocity

v w :

suction or injection velocity

x :

coordinate measured along surface

y :

coordinate normal to surface

α :

thickness of wall parameter

η :

similarity variable

ε :

thermal conductivity parameter

κ :

fluid thermal conductivity

κ eff :

effective thermal conductivity

ψ :

stream velocity function

ρ :

fluid density

σ*:

Stefan-Boltzmann constant

θ :

dimensionless temperature

μ :

fluid viscosity

ν :

kinematic viscosity of the fluid

∞:

free stream condition

w:

condition at surface

References

  1. Crane, L. J. Flow past a stretching plate. Zeitschrift für angewandte Mathematik und Physik, 21, 645–647 (1970)

    Article  Google Scholar 

  2. Gupta, P. S. and Gupta, A. S. Heat and mass transfer on a stretching sheet with suction or blowing. The Canadian Journal of Chemical Engineering, 55(6), 744–746 (1979)

    Article  Google Scholar 

  3. Soundalgekar, V. M. and Ramana, T. V. Heat transfer past a continuous moving plate with variable temperature. Wärme-und Stöffübertrag-ung, 14, 91–93 (1980)

    Article  Google Scholar 

  4. Grubka, L. J. and Bobba, K. M. Heat transfer characteristics of a continuous stretching surface with variable temperature. Heat and Mass Transfer, 107, 248–250 (1985)

    Google Scholar 

  5. Chen, C. K. and Char, M. Heat transfer on a continuous stretching surface with suction or blowing. Journal of Mathematical Analysis and Applications, 35, 568–580 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hayat, T., Abbas, Z., and Javed, T. Mixed convection flow of a micropolar fluid over a non-linearly stretching sheet. Physics Letters A, 372, 637–647 (2008)

    Article  MATH  Google Scholar 

  7. Cheng, P. and Minkowycz, W. J. Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike. Journal of Geophysical Research: Atmospheres, 82, 2040–2044 (1977)

    Article  Google Scholar 

  8. Elbashbeshy, E. M. A. and Bazid, M. A. A. Heat transfer in a porous medium over a stretching surface with internal heat generation and suction or injection. Applied Mathematics and Computation, 158, 799–807 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cortell, R. Flow and heat transfer of a fluid through a porous medium over a stretching surface with internal heat generation/absorption and suction/blowing. Fluid Dynamics Research, 37, 231–245 (2005)

    Article  MATH  Google Scholar 

  10. Hayat, T., Abbas, Z., Pop, I., and Asghar, S. Effects of radiation and magnetic field on the mixed convection stagnation-point flow over a vertical stretching sheet in a porous medium. International Journal of Heat and Mass Transfer, 53, 466–474 (2010)

    Article  MATH  Google Scholar 

  11. Hossain, M. A., Alim, M. A., and Rees, D. The effect of radiation on free convection from a porous vertical plate. International Journal of Heat and Mass Transfer, 42, 181–191 (1999)

    Article  MATH  Google Scholar 

  12. Hossain, M. A., Khanfer, K., and Vafai, K. The effect of radiation on free convection flow of fluid with variable viscosity from a porous vertical plate. International Journal of Thermal Sciences, 40, 115–124 (2001)

    Article  Google Scholar 

  13. Elbashbeshy, E. M. A. and Dimian, M. F. Effects of radiation on the flow and heat transfer over a wedge with variable viscosity. Applied Mathematics and Computation, 132, 445–454 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Abel, M. S. and Mahesha, N. Heat transfer in MHD viscoelastic fluid flow over a stretching sheet with variable thermal conductivity, non-uniform heat source and radiation. Applied Mathematical Modelling, 32, 1965–1983 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bataller, R. C. Radiation effects in the Blasius flow. Applied Mathematics and Computation, 198, 333–338 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Fang, T., Zhang, J., and Zhong, Y. Boundary layer flow over a stretching sheet with variable thickness. Applied Mathematics and Computation, 218, 7241–7252 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Khader, M. M. On the numerical solutions for the fractional diffusion equation. Communications in Nonlinear Science and Numerical Simulation, 16, 2535–2542 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Khader, M. M. and Megahed, A. M. Numerical simulation using the finite difference method for the flow and heat transfer in a thin liquid film over an unsteady stretching sheet in a saturated porous medium in the presence of thermal radiation. Journal of King Saud University Engineering Sciences, 25, 29–34 (2013)

    Article  Google Scholar 

  19. Khader, M. M., Talaat, S. D., and Hendy, A. S. A computational matrix method for solving systems of high order fractional differential equations. Applied Mathematical Modelling, 37, 4035–4050 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sweilam, N. H., Khader, M. M., and Adel, M. Numerical simulation of fractional Cable equation of spiny neuronal dendrites. Journal of Advanced Research, 1(1), 59–67 (2013)

    Google Scholar 

  21. Sweilam, N. H., Khader, M. M., and Nagy, A. M. Numerical solution of two-sided space fractional wave equation using finite difference method. Computational and Applied Mathematics, 235, 2832–2841 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Khader, M. M., Sweilam, N. H., and Mahdy, A. M. S. Numerical study for the fractional differential equations generated by optimization problem using Chebyshev collocation method and FDM. Applied Mathematics Information Science, 7(5), 2011–2018 (2013)

    Article  MathSciNet  Google Scholar 

  23. Zhou, J. K. Differential Transformation and Its Application for Electrical Circuits, Huazhong University Press, Wuhan (1986)

    Google Scholar 

  24. Borhanifar, A. and Abazari, R. Numerical study of nonlinear Schrödinger and coupled Schrödinger equations by differential transformation method. Optics Communications, 283, 2026–2031 (2010)

    Article  MATH  Google Scholar 

  25. Abazari, R. and Borhanifar, A. Numerical study of the solution of the Burger and coupled Burger’s equations by a differential transformation method. Computers and Mathematics with Applications, 59, 2711–2722 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Bildik, N. and Konuralp, A. The use of variational iteration method, differential transform method and Adomian decomposition method for solving different type of nonlinear partial differential equations. International Journal of Nonlinear Scince and Numerical Simulation, 7(1), 65–70 (2006)

    MathSciNet  MATH  Google Scholar 

  27. Jang, M. J., Chen, C. L., and Liy, Y. C. Two-dimensional differential transform for partial differential equations. Applied Mathematics and Computation, 121, 261–270 (2001)

    Article  MathSciNet  Google Scholar 

  28. Ozdemir, O. and Kaya, M. O. Flapwise bending vibration analysis of a rotating tapered cantilever Bernoulli-Euler beam by differential transform method. Journal of Sound and Vibration, 289, 413–420 (2006)

    Article  MATH  Google Scholar 

  29. Arikoglu, A. and Ozkol, I. Solutions of integral and integro-differential equation systems by using differential transform method. Computers and Mathematics with Applications, 56, 2411–2417 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Arikoglu, A. and Ozkol, I. Solution of difference equations by using differential transform method. Applied Mathematics and Computation, 174, 1216–1228 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Raptis, A. Flow of a micropolar fluid past a continuously moving plate by the presence of radiation. International Journal of Heat and Mass Transfer, 41, 2865–2866 (1998)

    Article  MATH  Google Scholar 

  32. Raptis, A. Radiation and viscoelastic flow. International Communications in Heat and Mass Transfer, 26, 889–895 (1999)

    Article  Google Scholar 

  33. Chiam, T. C. Magnetohydrodynamic heat transfer over a non-isothermal stretching sheet. Acta Mechanica, 122, 169–179 (1997)

    Article  MATH  Google Scholar 

  34. Mahmoud, M. A. A. and Megahed, A. M. MHD flow and heat transfer in a non-Newtonian liquid film over an unsteady stretching sheet with variable fluid properties. Canadian Journal of Physics, 87, 1065–1071 (2009)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, 11432, Saudi Arabia

    M. M. Khader

  2. Department of Mathematics, Faculty of Science, Benha University, Benha, 13511, Egypt

    M. M. Khader & A. M. Megahed

Authors
  1. M. M. Khader
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. M. Megahed
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. M. Khader.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khader, M.M., Megahed, A.M. Differential transformation method for studying flow and heat transfer due to stretching sheet embedded in porous medium with variable thickness, variable thermal conductivity, and thermal radiation. Appl. Math. Mech.-Engl. Ed. 35, 1387–1400 (2014). https://doi.org/10.1007/s10483-014-1870-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-014-1870-7

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation